#72314175

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	#7623363	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ #20475 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2819 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2,5,8)(3,7,4,6)(9,10)(11,20)(12,17)(13,18)(14,16)(15,19)(21,23)(22,36)(24,33)(25,32)(26,35)(27,31)(28,29)(30,34)(37,39)(38,40);(1,3,5,4)(2,6,8,7)(9,10)(11,20)(12,23,26,31)(13,28)(14,16)(15,19)(17,21,35,27)(18,29)(22,32)(24,39,30,40)(25,36)(33,37,34,38);(1,3,5,4)(2,7,8,6)(11,14)(12,24,26,30)(13,25)(16,20)(17,33,35,34)(18,32)(21,37,27,38)(22,29)(23,39,31,40)(28,36);(1,4,5,3)(2,7,8,6)(9,11)(10,16)(12,24)(13,29)(14,15)(17,34)(19,20)(21,37)(23,40)(26,30)(27,38)(31,39)(32,36)(33,35);(2,8)(3,4)(9,12,25,38,11,21,13,30,15,26,22,37,14,27,29,24)(10,17,32,40,20,23,18,34,19,35,36,39,16,31,28,33)

Polycyclic presentation:

2^10|a^2=h,b^2=f,c^2=f,d^2=h,e^2=fgi,g^2=i,i^2=j,b^a=bh,d^a=dhj,d^b=dj,d^c=dj,e^a=eh,e^b=ehij,e^c=eg,e^d=egi,f^e=fj,g^b=gj,g^c=gi,g^d=gij,g^e=gj,i^c=ij,i^d=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314251, #72314441, #72314604
maximal normal subgroup of	#72314251, #72314441, #72314604

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.8);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 5, F.6*F.7*F.9);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);